Note that the dressed electric charges (REF ) could acquire renormalizations different for each particle and chirality but, as we will see below,
e.m. chiral gauge invariance ensures that this is not the case; the renormalizations cancel out and the bare and dressed charge are equal.
While the symmetry (REF ) dictates the form of the e.m. current (REF ), this is not true for the axial current
(REF ); its definition is dictated by the fact that it must reduce to the standard definition of the chiral current in the continuum limit
and by the fact that the charges
carried by the axial current are the same as the one associated to the e.m. current. This is obtained
by properly tuning the normalizations
\(Z^5_{ i ,s}\)  in (REF ) so that (REF ) holds.
The lack of protection of the charges
associated to the chiral current is due to the lattice breaking
of chiral symmetry, but is even true in perturbative QFT with dimensional regularization, see [1]}; the absence of corrections in the anomaly is true
only provided that the correct normalization for the axial current is chosen.
With condition (REF ), one can define the chiral currents as \({1\over 2}(\widehat{j}^{e.m.}_{\mu , i,p}\pm \widehat{j}^{5}_{\mu , i,p}) \) . Similarly in (REF ) we
impose that the charges associated to the \(L\) -handed component of the \(W\)  current are equal, and
the \(R\) -handed component is vanishing.
While the first two conditions are imposed for physical reasons,
the condition 3 is just for technical convenience; the dressed and bare masses are different, and fixing the value of the dressed mass
equal to the non interacting one is somewhat convenient. In particular, the dressed mass of fermions is assumed vanishing, but of course
all the analysis could be repeated choosing any dressed mass.
